Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:

import java.util.Scanner;

public class Main {

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();

        double[] probs = new double[n];
        for (int i = 0; i < n; i++) {
            probs[i] = sc.nextDouble();
        }

        double[][] dp = new double[n + 1][n + 1];
        dp[0][0] = 1.0;

        for (int i = 1; i <= n; i++) {
            for (int j = 0; j <= n; j++) {

                double p1 = dp[i-1][j];
                double p2 = (j > 0) ? dp[i-1][j-1] : 0;
                double p = probs[i - 1];
                dp[i][j] = p1 * (1.0 - p) + p2 * p;
            }
        }

        double result = 0.0;
        for (int i = n/2 + 1; i <= n; i++) {
            result += dp[n][i];
        }

        System.out.println(result);

    }
}